ABSTRACT This study is devoted to unbiased motion of a point Brownian particle that escapes from a spherical cavity through a round hole. Effective one-dimensional description in terms of the generalized Fick-Jacobs equation is used to derive a formula which gives the mean first-passage time as a function of the geometric parameters for any value of a, where a is the hole’s radius. This is our main result and is given in equation (19). This result is a generalization of the Hill’s formula, which is restricted to small values of a.
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nullM. Vazquez and L. Dagdug, "Unbiased Diffusion to Escape through Small Windows: Assessing the Applicability of the Reduction to Effective One-Dimension Description in a Spherical Cavity," Journal of Modern Physics, Vol. 2 No. 4, 2011, pp. 284-288. doi: 10.4236/jmp.2011.24037.
 S. Redner, “A Guide to First Passage Time Processes,” Cambridge University Press, 2001.
 P. H?nggi, P. Talkner and M. Borkovec, “Reaction-rate Theory: Fifty Years after Kramers,” Reviews of Modern Physics, Vol. 62, No. 2, pp. 251-341.
 M. Coppey, O. Bénichou, R. Voituriez and M. Moreau, “Kinetics of Target Site Localization of a Protein on DNA: A Stochastic Approach,” Biophysical Journal, Vol. 87, No. 3, pp. 1640-1649.
 O. Bénichou, M. Coppey, M. Moreau, P. H. Suet and R. Voituriez, “Optimal Search Strategies for Hidden Targets,” Physical Review Letters, Vol. 94, No. 19, pp. 198101 (1-4).
 L. Gallos, C. Song, S. Havlin and H. A. Makse, “Scaling Theory of Transport in Complex Biological Networks,” Proceedings of the National Academy of Sciences U. S. A., Vol. 104, No. 19, pp. 7746-7751.
 D. Holcman and Z. Schuss, “Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane,” Journal of Statistical Physics, Vol. 117, No. 5--6, pp. 975-1014.
 O. Bénichou, and R. Voituriez, “Narrow-Escape Time Problem: Time Needed for a Particle to Exit a Confining Domain through a Small Window,” Physical Review Letters, Vol. 100, pp. 168105(1-4).
 Z. Schuss, A. Singer and D. Holcman, “The Narrow Escape Problem for Diffusion in Cellular Microdomains,” Proceedings of the National Academy of Sciences U. S. A., Vol. 104, No. 41, pp. 16098-16103.
 S. W. Cowan, T. Schirmer, G. Rummel, M. Steiert, R. Ghosh, R. A. Pauptit, J. N. Jansonius, and J. P. Rosenbusch, Nature, Vol. 358, pp. 727. doi:10.1038/358727a0
 L. Z. Song, M. R. Hobaugh, C. Shustak, S. Cheley, H. Bayley and J. E. Gouaux, Science, Vol. 274, 1996, pp. 1859. doi:10.1126/science.274.5294.1859
 M. Gershow and J. A. Golovchenko, “Recapturing and Trapping Single Molecules with a Solid-state Nanopore,” Nature Nanotechnology, Vol. 2, pp. 775-779.
 L. T. Sexton, L. P. Horne, S. A. Sherrill, G. W. Bishop, L. A. Baker and C. R. Martin, “Resistive-Pulse Studies of Proteins and Protein/Antibody Complexes Using a Conical Nanotube Sensor,” Journal of the American Chemical Society, Vol. 129, No. 43, pp. 13144-13152.
 I. D. Kosinska, I. Goychuk, M. Kostur, G. Schmidt and P. H?nggi, “Rectification in Synthetic Conical Nanopores: A One-dimensional Poisson-Nernst-Planck Model,” Physical Review E, Vol. 77, No. 3, pp. 031131.
 J. K?rger and D. M. Ruthven, “Diffusion in Zeolites and Other Microporous Solids,” Wiley, New York, 1992.
 R. A. Siegel, “Theoretical Analysis of Inward Hemispheric Release above and below Drug Solubility,” Journal of Controlled Release, Vol. 69, No. 1, pp. 109-126.
 N. F. Sheppard, D. J. Mears, and S. W. Straka, “Micromachined Silicon Structures for Modelling Polymer Matrix Controlled Release Systems ,” Journal of Controlled Release, Vol. 42, No. 1, pp. 15-24.
 M. H. Jacobs, “Diffusion Processes,” Springer, New York, 1967.
 R. Zwanzig, “Diffusion past an Entropy Barrier,” Journal of Physical Chemistry, Vol. 96, No. 10, pp. 3926-3930.
 D. Reguera and J. M. Rub, “Kinetic Equations for Diffusion in the Presence of Entropic Barriers,” Physical Review E, Vol. 64, No. 6, pp. 061106(1-8).
 A. M. Berezhkovskii, M. A. Pustovoit and S. M. Bezrukov, “Diffusion in a Tube of Varying Cross Section: Numerical Study of Reduction to Effective One-Dimensional Description,” Journal of Chemical Physics, Vol. 126, No. 13, pp. 134706(1-5).
 P. Kalinay, “Mapping of Forced Diffusion in Quasi-One- Dimensional Systems,” Physical Review E, Vol. 80, No. 3, pp. 031106(1-10).
 R. M. Bradley, “Diffusion in a Two-Dimensional Channel with Curved Midline and Varying Width: Reduction to an Effective One-Dimensional Description,” Physical Review E, Vol. 80, No. 6, pp. 061142(1-7).
 P. S. Burada and G. Schmid, “Steering the Potential Barriers: Entropic to Energetic,” Physical Review E, Vol. 82, No. 5, pp. 051128(1-6).
 I. V. Grigoriev, Yu. A. Makhnovskii, A. M. Berezhkovskii and V. Yu. Zitserman, “Kinetics of Escape through a Small Hole,” Journal of Chemical Physics, Vol. 116, No. 22, pp. 9574-9577.